Generalizations of the Strong Arnold Property and the Inverse Eigenvalue Problem of a Graph

Transcription

1 Generalizations of the and the Inverse Eigenvalue Problem of a Graph Iowa State University and American Institute of Mathematics MM Joint Mathematics Meetings Atlanta, GA, January 7, 2017 Joint work with W. Barrett, S. Butler, S. Fallat, H.T. Hall, J. C.-H. Lin, B.L. Shader, and M. Young

2 Inverse Eigenvalue Problem of a Graph Matrices are real and symmetric unless otherwise stated. All graphs are simple, undirected and finite. The graph G(A) = (V, E) of n n matrix A is G = (V, E) where V = {1,..., n}, E = {ij : a ij 0 and i j}. Diagonal of A is ignored. The family of symmetric matrices described by a graph G is MM S(G) = {A : A T = A and G(A) = G}. The Inverse Eigenvalue Problem of graph G is to determine all possible spectra (multisets of eigenvalues) of matrices in S(G).

3 Maximum multiplicity and minimum rank Due to the difficulty of the Inverse Eigenvalue Problem, a simpler form called the maximum multiplicity or minimum rank problem has been studied. The maximum multiplicity or maximum nullity of graph G is M(G) = max{mult A (λ) : A S(G), λ spec(a)}. The minimum rank of graph G is By using nullity, mr(g) = min rank A. A S(G) M(G) + mr(g) = G. MM The Maximum Multiplicity Problem (or Minimum Rank Problem) for a graph G is to determine M(G) (or mr(g)).

4 Matrix A has the (SAP) if the zero matrix is the only real symmetric matrix X such that A X = 0 and I n X = 0 AX = 0. A Colin de Verdière type parameter of graph G is the maximum nullity of (real symmetric) matrices A having G(A) = G, satisfying SAP, and possibly other. ξ(g) = max{null(a) : G(A) = G and A has SAP}. MM ξ(g) M(G).

5 s and minors A graph H = (V H, E H ) is a subgraph of G = (V G, E G ) (H G) if V H V G and E H E G. H is a minor of G (H G) if H can be obtained from G by performing a sequence of deletions of edges, deletions of isolated vertices, and/or contractions of edges. A subgraph is a minor. A graph parameter ζ is minor monotone if H G ζ(h) ζ(g). [Barioli, Fallat, Hogben; 2005] ξ is minor monotone. MM

6 SAP, manifolds, and generalizations Matrix A has SAP if and only if the constant rank manifold and the constant pattern manifold S(G(A)) intersect transversally at A. SAP is used to guarantee minor. SAP and the Colin de Verdière type parameters are easier to compute than other minor monotone parameters related to maximum multiplicity, such as minor monotone floor of maximum multiplicity. We consider the transverse intersection of other relevant manifolds to obtain subgraph and more generally minor of additional spectral. The can be applied to the inverse eigenvalue problem. MM

7 Strong Spectral () Λ is a multiset of real numbers of cardinality n. The set of all n n real symmetric matrices with spectrum Λ is denoted by E Λ. E spec(a) is the set of all symmetric matrices cospectral with A. It is well known that E Λ is a manifold. The commutator AB BA of two matrices is denoted by [A, B]. Matrix A has the Strong Spectral () if the zero matrix is the only symmetric matrix X satisfying A X = O, and I X = O [A, X ] = O. Since AX = O implies [A, X ] = O, if A has the, then A has the SAP. MM

8 Example: show a matrix has Example A = X T = X, A X = 0, I X = O imply X = 0 0 u v 0 u 0 w 0 v w 0. [A, X ] = 0 implies that X has all row sums and column sums equal to zero, which in turn implies X = O. Thus, A has. MM

9 Ordered multiplicity lists Suppose the distinct eigenvalues of A are λ 1 < λ 2 < < λ q and the multiplicity of these eigenvalues are m 1, m 2,..., m q. The ordered multiplicity list of A is m(a) = (m 1, m 2,..., m q ). Inverse Ordered Multiplicity List Problem: Given a graph G, determine which ordered multiplicity lists arise among the matrices in S(G). The Inverse Ordered Multiplicity List Problem lies in between the Inverse Eigenvalue Problem and the Maximum Multiplicity Problem. MM

10 Strong Multiplicity () m = (m 1,..., m q ) is an ordered list of positive integers with m 1 + m m q = n. U m is the set of all symmetric matrices whose ordered multiplicity list is m. U m(a) is the set of symmetric matrices has the same ordered multiplicity list as A. It follows from results of Arnold that U m is a manifold. The n n symmetric matrix A satisfies the Strong Multiplicity () provided no nonzero symmetric matrix X satisfies A X = O, and I X = O, [A, X ] = O, and tr(a i X ) = 0 for i = 0,..., n 1. implies. MM

11 ,, and manifolds Theorem (BFHHLS; 2015+) Matrix A has the if and only if the manifolds S(G(A)) and E spec(a) intersect transversally at A. Matrix A has the if and only if the manifolds S(G(A)) and U m(a) intersect transversally at A. MM

12 Theorem (BFHHLS15+) G is a graph of order n, Ĝ is of order ˆn, G Ĝ. If A S(G) has, then there exists  S(Ĝ) with such that spec(â) = spec(a) Λ where Λ is any set of distinct real numbers such that spec(a) Λ =. If A S(G) has, then there exists  S(Ĝ) with such that m(â) is obtained from m(a) by extending with 1s in any positions. MM

13 Diagonal and block diagonal matrices and the A diagonal matrix D with distinct eigenvalues has : DX = XD implies all off-diagonal entries of X are zero. For any graph on n vertices and any set Λ of n distinct real numbers, there is a realization A that has and spec(a) = Λ. The existence of such a matrix was proved in [Monfarad, Shader; 2015] via a different method. Theorem (BFHHLS; 2015+) MM Let A 1 and A 2 be symmetric matrices. Then A := A 1 A 2 has the (respectively, ) if and only if both A 1 and A 2 have the (respectively, ) and spec(a 1 ) spec(a 2 ) =.

14 (3, 1) (1, 3) K4 (4, 1) (1, 4) K5 with without (2, 2) (2, 1, 1) (1, 1, 2) (1, 2, 1) (1, 1, 1, 1) (2, 1) (1, 2) (1, 1, 1) Dmnd C4 Paw K1,3 P4 K3 P3 (3, 2) (2, 3) (2, 1, 2) (3, 1, 1) L(4, 1) (1, 1, 3) (3, 1, 1) Bfly (1, 3, 1) (1, 1, 3) (1, 3, 1) K2,3 T5 (2, 2, 1) (1, 2, 2) FHs (K4)e Bnr Dart Kite C5 Hs K5 e W5 Gem Camp (1, 3, 1) K1,4 MM (1, 1) K2 (2, 1, 1, 1) (1, 1, 1, 2) L(3, 2) Bull (1, 2, 1, 1) (1, 1, 2, 1) S(2, 1, 1) (1) K1 (1, 1, 1, 1, 1) P5

15 order 5 The diagram shows the connected graphs of order at most 5 with their ordered multiplicity lists. If a box is joined to another box by a line then the graphs in the upper box can realize every ordered multiplicity list of the graphs in the lower box (including other boxes below connected with lines to lower boxes). Every ordered multiplicity list is spectrally arbitrary for the graphs that attain it. MM Theorem (BBFHHLSY; 2016+) The diagram is correct (it lists all ordered multiplicity lists for each connected graph of order n 5).

16 Minimum number of distinct eigenvalues For a matrix A, q(a) is the number of distinct eigenvalues of A. For a graph G, the minimum number of distinct eigenvalues of G is q(g) := min{q(a) : A S(G)}. Minimum Number of Distinct Eigenvalues Problem: Determine q(g). MM

17 Minimum number of distinct eigenvalues q (G) := min{q(a) : A S(G) and A has } q (G) := min{q(a) : A S(G) and A has }. q(g) q (G) q (G). Theorem (BFHHLS; 2015+) If G is a subgraph of Ĝ, G = n, and Ĝ = ˆn, then: MM q(ĝ) q (Ĝ) ˆn (n q (G)). q(ĝ) q (Ĝ) ˆn (n q (G)).

18 High minimum number of distinct eigenvalues Proposition Let G be a graph. Then the following are equivalent: (a) q(g) = G, (b) M(G) = 1, (c) G is a path. Theorem (BFHHLS; 2015+) A graph G has q(g) G 1 if and only if G is one of the following: (a) a path, (b) the disjoint union of a path and an isolated vertex, (c) a path with one leaf attached to an interior vertex, (d) a path with an extra edge joining two vertices at distance 2. MM

19 Proposition (BFHHLS; 2015+) Let G be one of the graphs H-tree, campstool, S(2, 2, 2)-tree, or 3-sun shown below. Then q (G) G 2. campstool H-tree Theorem (BFHHLS; 2015+) 3-sun Let C n be the cycle on n 3 vertices. Then n q (C n ) =. 2 S(2,2,2) MM

20 Monotonicity Theorem Theorem (BBFHHJSY; 2016+) Suppose G is a minor of H obtained by contraction of r edges, deletion of s vertices, and deletion of any number of edges, and A S(G). If A has and m(a) = (m 1,..., m t ), then there is a matrix A S(H) having with m(a ) obtained from m(a) by adding r + s ones, with at most s of these between m 1 and m t. If in addition A has the, then A can be chosen to have the, spec(a) spec(a ), the remaining eigenvalues are simple, and s of the additional simple eigenvalues can be chosen to have any values (including between λ min(a) and λ max(a) ). MM

21 Forbidden minors for two multiple eigenvalues. K 3 U K 3. K 3 U K 1,3. K 1,3 U K 1,3 C 4 Campstool H-tree MM 3-sun K 1,6 S(2,1,1,1,1) S(2,2,1,1) S(2,2,2)

22 Matrix Liberation and Augmentation Given a matrix B, the Matrix Liberation Lemma gives conditions under which edges can be added to G = G(B) to obtain graph Ĝ while guaranteeing the existence of a matrix ˆB S(Ĝ) with the, or SAP and spec( ˆB) = spec(b), m( ˆB) = m(b), or rank ˆB = rank B. Given a matrix B with the and λ spec(b), the Augmentation Lemma gives conditions on the eigenvectors of B for λ that allow another vertex and certain edges to be added to G = G(B) to obtain graph Ĝ while guaranteeing the existence of a matrix ˆB S(Ĝ) with the and spec( ˆB) = spec(b) {λ}. An application of the Augmentation Lemma shows that for a cycle C n, any ordered ordered multiplicity list that has one 2 and all the other entries 1 (in any order) is spectrally arbitrary with the. MM

23 Bifurcation Theorem (BBFHHLSY; 2016+) Suppose A S(G) has the and m(a) = (m 1, m 2,..., m q ). If n 1 + n , n p = m j, then there exists B S(G) having the and m(b) = (m 1,..., m j 1, n 1,..., n p, m j+1,..., m q ). MM

24 International Linear Algebra Society Conference MM Ames, Iowa, USA, July 24-28,

25 A graph and ordered multiplicity list m such that there exists B S(G) with m(b) = m and B has the, but no matrix A S(G) with m(a) = m has the : MM m = (3, 5, 4)